Should we devote our time to opposing the teaching of eternal hell? Never build an outreach on a negative that you're going to spend all of your time refuting this or that 'cause then you fall right into the hands of the critics that what you share does nothing to draw people to Christ. Emphasize the finished work of Christ and the supremacy of it over all other theories of justice and worthiness. That's what St. Paul did in handling the circumcision and other points of the Judaizers. Otherwise, a couple thousand years later we'd have 13 to 14 documents in our New Testament that would have no baring to our lives today 'cause they'd be exclusively about refuting the doctrine of circumcision. What you want to do is to always deal with both the present and the future and to build something that's relevant for the future even when the teaching of Eternal Hell has been abolished from the Church.
Wonderful scholars have already produced a legacy of writings that refute the Eternal Hell nonsense and we can point people in the direction of those resources and how to get them through Abebooks.com, concordant.org, tentmaker.org, etc. But focus on the fact that the most holy has been anointed and everlasting righteousness has been brought in according to Daniel's prophecy, and that as St. Paul put it "if in this life only we have hope, then we're of all men miserable." Keep on and on and on with the passages about the Blood of Christ speaking better things than the blood of Abel in the book of Hebrews, how it was one sacrifice [for all] to the close of the age of God's dealings with the righteous and the wicked, and how Ephesians 2:13 says that His Blood is what brings us near rather than our self righteousness.
Yes, Acts 17, Acts 20, and a few other passages do emphasize repentance and passages like Romans 10 and 2Corinthians 5 say that our real salvation begins when we call/cry out to Him, but God has already written the check in the Blood of His own Son to cover the cost that's ever been incurred by the sin of anyone and God's not accepting our own eternal punishment as a sufficient replacement in the bank of heaven for our sins instead of the Blood of Christ. None of us are our own, but all of us have been bought with a price because Jesus Christ is the ransom for all to be testified in due time, therefore we're to glorify God in our spirit, soul, and body and cleanse ourselves of all filthiness of flesh and spirit perfecting holiness in the fear of God. But God is not pleased in the death of the wicked, but in the recovery of the wicked from their death in trespasses.
It's to those that were dead in trespasses that He's ordained that in the ages to come that He'd show them the exceeding riches of His grace in His kindness towards them through Christ Jesus' High Priestly ministry. Israel in the Old Testament wasn't judged on their own righteousness, but upon the righteousness of their high priest. Likewise today, the world isn't judged on their own righteousness, but upon the righteousness of their High Priest. God's counsel shall stand and He will do absolutely all of His pleasure and He's sworn the oath that every knee will bow and every tongue will confess that Jesus Christ is Lord and that in His Lordship they have righteousness and strength. Be a peddler of the Cross and of nothing else and let the chips fall where they will when it comes to the New Testament pattern of emphasizing the Gospel of Grace rather than all of the other minutia that it's easy to devote the rest of your time on earth to.